| L(s) = 1 | − 9·4-s + 5·9-s − 52·11-s + 17·16-s + 252·19-s + 164·29-s + 392·31-s − 45·36-s + 672·41-s + 468·44-s + 517·49-s + 588·59-s − 112·61-s + 423·64-s + 18·71-s − 2.26e3·76-s − 2.60e3·79-s − 704·81-s + 2.38e3·89-s − 260·99-s + 840·101-s − 746·109-s − 1.47e3·116-s − 634·121-s − 3.52e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 9/8·4-s + 5/27·9-s − 1.42·11-s + 0.265·16-s + 3.04·19-s + 1.05·29-s + 2.27·31-s − 0.208·36-s + 2.55·41-s + 1.60·44-s + 1.50·49-s + 1.29·59-s − 0.235·61-s + 0.826·64-s + 0.0300·71-s − 3.42·76-s − 3.71·79-s − 0.965·81-s + 2.83·89-s − 0.263·99-s + 0.827·101-s − 0.655·109-s − 1.18·116-s − 0.476·121-s − 2.55·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.015782853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.015782853\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 9 T^{2} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 517 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 26 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3897 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 15118 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 82 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 196 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 84145 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 336 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 118613 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 196621 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 111130 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 294 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 56 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 373042 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 768430 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1304 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1048710 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1190 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1820446 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.63859853902766958533805296758, −10.90218250986425820185181797218, −10.24718905026366164739595548211, −9.881911178675541591343204151889, −9.808333387640733161149352267680, −8.860499942603873597129464957842, −8.852258217999307934659445328819, −8.009612631885813263248647149222, −7.65081380338706491375045999783, −7.32972379741439318663941514963, −6.57826418682071844389214211422, −5.67191848100360932297693639314, −5.55575037216678797525972358326, −4.76822272774925217182723638311, −4.53025110011016157091247341417, −3.74781814982312823176529748093, −2.83163346522983633259291050793, −2.63579099108165727178657843341, −1.07034761222790638200765200751, −0.65976476164030330908614974277,
0.65976476164030330908614974277, 1.07034761222790638200765200751, 2.63579099108165727178657843341, 2.83163346522983633259291050793, 3.74781814982312823176529748093, 4.53025110011016157091247341417, 4.76822272774925217182723638311, 5.55575037216678797525972358326, 5.67191848100360932297693639314, 6.57826418682071844389214211422, 7.32972379741439318663941514963, 7.65081380338706491375045999783, 8.009612631885813263248647149222, 8.852258217999307934659445328819, 8.860499942603873597129464957842, 9.808333387640733161149352267680, 9.881911178675541591343204151889, 10.24718905026366164739595548211, 10.90218250986425820185181797218, 11.63859853902766958533805296758
